THE CRYSTALLOGRAPHY OF THE CUBIC TO ORTHORHOMBIC
TRANSFORMATION IN THE ALLOY AuCu*
R. SMITH? and J. S. BOWLESZ
The habit plane, orientation relationships and shape change associated with the transformation to the ordered orthorhombic phase in the alloy CuAu have been measured and are shown to conform with the phenomenological theory of martensitic transformations.‘2-4’ The agreement with the theory is systematically better when some dilation of the habit plane is permitted. It has been found that groups of four plates having an irrational habit close to (1 10}c develop into compound plates parallel to { 110)~.
L’ASPECT CRISTALLOGRAPHIQUE DE LA TRANSFORMATION DU SYSTEME
CUBIQUE EN SYSTEME ORTHORHOMBIQUE DANS L’ALLIAGE Au-Cu
Les auteurs ant determine le plan d’habitat, les relations d’orientation et la modification de forme, qui sont associes B la formation de la phase orthorhombique ordonnee dans l’alliage Cu-Au.
11s montrent que ces relations sont conformes B la theorie phenomenologique des transformations martensitiques.‘2-*’ 11s constatent que l’accord avec la theorie est systematiquement meilleur lorsque l’on autorise une oertaine dilatation du plan d’habitat. 11s montrent que des groupes de quatre plaques ayant un habitat irrationnel proche de (110)~ se developpent en plaques composees paralleles L {llO}c.
KRISTALLOGRAPHIE DER KUBISCH-ORTHORHOMBISCHEN UMWANDLUNG
DER LEGIERUNG AuCu
Habitusebene, Orientierungsbeziehungen und GestaltBnderung, die mit der Umwandlung der Legienmg CuAu in die geordnete orthorhombische Phase verbunden sind, wurden gemessen; sie stimmen mit der phknomenologischen Theorie martensitischer Umwandlungen(2-4~ iibersin. Die Ubereinstim- mung mit der Theorie wird systematisch verbessert, wenn man eine gewisse Dilation der Habitusebene zul&Dt. Gruppen von vier platten, die einen irrationalen Habitus nahe an {110}, haben, werden su zusammengesetzten Platten parallel zu {llo},.
1. INTRODUCTION
Investigation of the precipitation of the phase CuBe
during ageing of copper-beryllium alloys(l) has shown
that the crystallography of this transformation is
consistent with the phenomenological theory of
martensitic transformations.(2-4) This result provides
some support for the idea that a common (dislocation)
mechanism may be involved both in martensitic
transformations and in the early stages of diffusion
controlled transformations. The present investigation
of the transformation in the alloy AuCu was under-
taken to test further the validity of this idea.
The ordering in AuCu and the accompanying
changes in lattice symmetry have been the subject of
numerous investigations.(5-g) These investigations
have shown that two closely related superlattices with
different symmetries can form from the disordered
face-centred cubic structure. When ordering occurs
below approximately 38O”C, the ordered phase has a
face-centred tetragonal structure in which the copper
and gold atoms occupy alternate planes normal to the
c axis. Between approximately 410°C and 380°C a
complex ordered structure forms which has a pseudo-
cell which is face-centred orthorhombic. This pseudo-
cell has a b/a ratio in the range 0.9982-0.997, but apart
from this it is similar in dimensions to the unit cell of
the ordered tetragonal structure. According to
Johansson and Lindec6), the complex ordered structure
has an orthorhombic unit cell comprising ten .of the
pseudo-cells side by side in the direction of the a axis;
the atomic planes perpendicular to the c axis are again
occupied alternately by copper and gold atoms but the
stacking order of the atoms changes over after five
pseudo-cells. The axial ratio c/b of both structures
decreases with decreasing temperature from about
0.939 at 410°C to about 0.926 at 300°C.
Isothermal transformation studies of ea,ch trans-
formation’g>lO) have established beyond doubt that the
rates are diffusion-controlled; typical C-shaped curves
are obtained if the times for different degrees of
transformation are plotted against the transformation
temperature. On the other hand, the microstructures
produced by these transformation@) are similar in
appearance to those produced by martensitic trans-
formations. Because of this similarity it seemed
probable that the transformations would produce
relief effects on polished surfaces. Preliminary experi-
ments showed that this was indeed the case, so that it
became of interest to discover whether the geo-
metrical features of these transformations conformed
with the phenomenological theory of martensitic
* Received April 15, 1959.
C&k N 3 ’ Victoria ormerly School of Metallurgy, University of Melbourne,
now Australian Atomic Energy Research E&blishme& Sutherland, N.S.W., Australia.
2 School of Metallurgy, University of New South Wales, Kensington, N.S.W., Australia.
ACTA METALLURGICA, VOL. 8, JULY 1960 405
1
406 ACTA METALLURGICA, VOL. 8, 1960
transformations. It was found that the cubic to
tetragonal transformation does not proceed by the
production of tetragonal crystals within the cubic
matrix so that it was not appropriate to attempt to
apply the theory to this transformation. The cubic to
orthorhombic transformation, on the other hand, was
found to proceed by the growth of orthorhombic
crystals at the expense of the cubic matrix and so
should conform with the theory. Accordingly,
measurements of lattice parameters, orientation
relationships, habit planes, and surface relief, were
made for the cubic or orthorhombic transformation.
The general characteristics of each of the trans-
formations are described in Section 2 and the detailed
analyses of the crystallography of the cubic to ortho-
rhombic transformation in Section 3. In Section 4 the
results of this investigation are compared with the
predictions of the theory.
2. GENERAL CHARACTERISTICS
(a) Experimental methods
The materials used were oxygen-free high con-
ductivity copper and “double-refined” gold, both of
99.99 wt.% purity; the copper contained 0.002 wt.%
of silver, oxygen and sulphur, and the main impurities
in the gold were silver and copper. Cylindrical speci-
mens (6 mm diameter by 6 cm long), of equiatomic
composition were prepared by induction melting in
evacuated silica capsules, and these specimens were
then converted to single crystals by the Bridgman
technique. Some of the single crystals were allowed
to cool to room temperature in the temperature
gradient to yield the tetragonal structure; others
were water-quenched from above 450°C to retain the
disordered cubic structure at room temperature.
To produce the orthorhombic structure, specimens
cut from the single crystals were heated for 1 hr
at 800°C and transformed isothermally at tempera-
tures in the range 389-405”C. The tetragonal
structure was studied mainly on specimens cut from
the single crystals which had been cooled in the
temperature gradient, but some observations were
also made on specimens heated for 1 hr at 8OO”C,
quenched to room temperature and then transformed
isothermally at temperatures in the range 250-350°C.
These heat treatments were usually carried out in a
salt bath, the specimens being contained in evacuated
silica capsules, but in a few cases specimens were heat
treated in the evacuated chamber of a hot stage
microscope and the progress of the transformation
followed visually.
For the study of relief effects, the specimens were
obtained initially in the disordered cubic condition,
and metallographically polished before heat-treatment.
A mechanical polish sufficed for this purpose, but for
the general examination of microstructures, specimens
were polished either electrolytically or by “polish-
attack”. The electrolyte consisted of hydrochloric
acid, nitric acid and water in the proportions 1: 3 : 4 by
volume, and polishing was carried out at 0.8 V.
A mixture of hydrochloric and chromic acids was used
in the polish-attack method.
(b) Experimental observations
Transformation in the temperature range 380-410°C
produced microstructures of the type illustrated
in Fig. 1. The plates in these microstructures were
observable either as relief effects or after polishing
and etching. When etched specimens were examined
in polarized light, a single system of fine parallel
sub-bands became visible in each plate (Figs. 2 and 3).
It was established by X-ray examination of partly
transformed specimens that the plates had the ortho-
rhombic structure and the surrounding matrix, the
FIG. 1. Relief effects produced after 18 hr at 400°C. x50
FIG. 2. Microstructure showing sub-bands, “diamond” figures and pseudo habit plane, in specimen transformed
18 hr at 400°C. Polarized light. x 150
SMITH AND BOWLES: TRANSFORMATION IN SuCu 407
FIG. 3. “Diamond” figures in specimen transformed 2 hr at 390°C. Polarized light. x 300
disordered face-centred cubic structure. This is
consistent with the results of Kallbach et aZ.(g) who
showed that the orthorhombic phase develops at the
expense of the cubic phase.
The plates produced during the early stages of the
transformation were almost invariably arranged in
groups of four and appeared on the surface being
examined as roughly diamond shaped figures. From
the form of these figures and the fact that habit plane
determinations showed that the group consisted of
four non-parallel variants of the same crystallographic
plane, it seems likely that the groups of plates had the
form of a pyramid. This suggests strongly that all four
plates were nucleated at a common point, the apex
of the pyramid.
Observation with the hot stage microscope showed
that, in all cases, transformation commenced by the
formation of isolated groups of small plates which
then grew slowly by lengthwise growth and, to a
lesser extent, by thickening of the plates. Occasionally,
a fully grown group of plates appeared suddenly, but
this can be attributed to the arrival at the surface of a
group of plates nucleated in the interior. In some
FIR. 5. Large numbers of small plates formed in 5 rnin at 3RO’C. x200
cases, when the plates had grown to a certain stage,
other plates grew uniformly along the four sides of the
diamond, and occasionally lengthwise extension
occurred by the formation of another set of plates.
This, together with the growth of the plates towards
the interior of the pyramid, led to the development
of long bands parallel to a pseudo habit plane (Fig. 4).
In agreement with the results of other workers,cgJO)
the kinetics were found to be typical of a diffusion-
controlled transformation, with the overall rate of
transformation being strongly temperature-dependent.
At 400°C there was an incubation period of approxi-
mately 15 hr, and the transformation was usually
completed in 25 hr. At 390°C transformation started
after about 1.5 hr and was complete in 4 hr. At 380°C
transformation was complete in 20 min. At higher
temperatures the rate of nucleation was small and a
relatively small number of well-developed plates was
produced (Fig. 2), but at lower temperatures nucleation
was rapid and a large number of small plates was
produced (Fig. 5).
Isothermal transformation to the tetragonal phase
at temperatures in the range %O-350°C yielded
FIG. 4. Specimen completely transformed after 4 hr at 390°C. Polarized light. x 100
FIG. 6. Single crystal cooled in temperature gradient, showing tetragonal phase with (101) twins and slip markings. x40
408 ACTA METALLURGICA, VOL. 8, 1960
microstructures of the type shown in Fig. 6, the
structure being observable either in relief or after
polishing and etching. Similar structures were
obtained when the single crystals were slowly cooled
in the temperature gradient, and in this case, the
formation of the structure was accompanied by
audible clicks and produced a macroscopic distortion
of the crystal. The microstructures consisted of one
or more families of wide bands within which were
numerous slip-markings. At first it .was thought that
the bands could be identified as plates of the ordered
tetragonal phase growing at the expense of a disordered
cubic matrix. However, X-ray examination of the
specimen illustrated in Fig. 6 showed that both the
bands and the continuous matrix in which they
appeared were ordered and tetragonal (a = 3.95 A,
c = 3.68 A). The bands were found to be (101) twins
of the tetragonal matrix, and an analysis of the relief
effects showed that the tilts produced on two surfaces
by the formation of the bands were those that would
be expected if the bands were formed by mechanical
twinning of the tetragonal matrix.
The tetragonal transformation was followed both on
the hot-stage microscope during cooling from 800°C
and also while heating specimens that had previously
been quenched; the rate of change of temperature in
each case was of the order of 50”C/min. These
experiments showed that during heating, the banded
structure formed at about 250°C and during cooling
at about 360°C and that in each case it formed in a
matter of seconds. The full-sized bands seemed to
appear instantaneously over the surface of the
specimen, but the relief effects were not well defined
at first, and some time elapsed before they became
fully developed. At no stage was there any indication
of a heterogeneous process in which plates of the
tetragonal phase were growing at the expense of a
cubic matrix.
Attempts were made to produce specimens contain-
ing both the cubic and tetragonal phases, by cooling
single crystals, 6 cm long, in a temperature gradient,
and quenching after transformation had begun at the
cooler end. Five specimens were produced in this
way, and each had a disordered cubic structure at one
end and an ordered tetragonal structure at the other
(Fig. 7). There was, however, no definite interface
between these two structures; instead there seemed
to be a zone where one structure changed gradually
into the other. A series of Laue and Weissenberg
patterns taken across the transition zone shown in
Fig. 7 showed that a single orientation of the dis-
ordered cubic structure existed from the untransformed
end of the crystal up to a point 2.5 mm from the end
of the tetragonal bands. Beyond this point and up to
point B (approximately 2 mm) two orientations
existed which had a maximum orientation difference
of IO. Close to point B these two orientations were
found to be slightly tetragonal (c/a = 0.99) with a
small degree of order. Beyond point B and up to
point C there was a marked increase in the degree of
order and a decrease in the axial ratio c/a, such that
at point C within the banded region the axial ratio
was approximately 0.939. Beyond point C the only
observable change was a slight decrease in the axial
ratio, which reached a constant value of approximately
0.935 at point D. From these results it is clear that, for the conditions
studied, the tetragonal phase does not form at the
expense of the cubic phase by the advancement of a
well-defined interface. The results are consistent with
the view that the bands are mechanical twins which
form after some degree of order and tetragonality
has developed.
3. CRYSTALLOGRAPHIC FEATURES OF THE
CUBIC TO ORTHORHOMBIC TRANSFORMATION
(a) Experimental methods
The detailed analysis of the crystallographic
features of the cubic to orthorhombic transformation
was carried out on a group of four well-developed
plates of the orthorhombic phase (Fig. 8). This group
of plates was produced in a specimen containing
50.2 at.% gold by heating to 800°C for 1 hr, trans-
forming isothermally at 400°C + 2°C for 42 hr and
then quenching to room temperature. Before this
heat-treatment the specimen had been obtained in the
disordered condition and one surface had been
polished.
The analysis of the homogeneous strain accompany-
ing the transformation was limited to the measurement
of the tilts produced by the four plates on the original
polished surface of the specimen. These measurements
were made by multiple-beam interferometry.
The habit planes of the plates and the plane of the
sub-bands were determined from trace measurements
on two surfaces. For this purpose, the specimen was
prepared as shown in Fig. 9 so that traces of each
plate were exposed on two surfaces. To minimize
errors due to the presence of a lineage structure in the
specimen, the trace measurements and the orientation
determinations were made on two small regions, one
for each pair of plates A, B and C, D at the positions
shown in Fig. 9. Since the sides of the plates were not
parallel, measurements were made of the traces on
both sides of each plate, and the mean values were
SMITH AND BOWLES: TRANSFORMATION IN AuCu 409
FIG. 7. Transition zone between the cubic structure and the twinned tetragonal structure in a single crystal quenched after partial transformation in a temperature
gradient. x 50
used for the habit plane determination. The crystal
orientations were determined by a rotating crystal
X-ray method in which two rotation patterns were
taken about known axes at right angles. The
orientation of particular planes could then be deter-
mined from the angles between the normals to the
planes and each of the rotation axes. The error in this
method was estimated to be less than 0.5”. The
measurements of the traces of the habit planes were
considered to be accurate to within kO.25” and those
of the sub-bands to within f2”.
The orientation relationships between the cubic and
orthorhombic phases were determined from oscillating
crystal X-ray photographs containing reflections from
both phases. The plates were examined two at a time
by masking off with a paste of litharge and glycerine,
all of the specimen surface except strips, 0.005 in.
wide, across the pairs of plates A,B and C,D. For
each pair of plates two oscillation photographs, about
axes at right angles, were taken on each of three
surfaces of the specimen. A typical pair of photo-
graphs showing a (200), reflection on the zero layer
line with the {200}, reflections clustered about it, is
reproduced in Fig. 10. In determining from such
patterns the relative positions of the (lOO}, and
{10010 axes, a correction had to be made in some
cases because of the combined effect of the divergence
of the beam and the separation of the plates, on the
spacing of the orthorhombic reflections.
The lattice parameter of the disordered face-centred cubic phase was determined from the reflections on
the zero layer line of a single crystal rotation pattern.
The parameters of the orthorhombic phase were
determined relative to this cubic parameter. For this
purpose several oscillating crystal patterns were
recorded on the same film so that reflections from
FIG. 8. Relief effects produced by the group of ortho- rhombic plates used in the crystallographic study. x 40
(400),, (400),, (040), and (004), were all obtained on
the zero layer line. The orthorhombic parameters
were calculated from the distances between the
cobalt Ku- reflections on such a film, and the known
cubic parameter.
A detailed account of these techniques is given
in Ref. (13).
FIG. 9. Sketch of specimen showing the traces of the four plates A, B, C and D on three surfaces.
k_J
0 /
FIa. 10. Oscillation patterns showing (200)~ reflections and associated group of (2OO)o reflections. CoKcc radia-
tion. x2.5
ACTA METALLURGICA, VOL. 8, 1960
Fm. Il. Stereographic projection showing the orienta- tion relative to the cubic axes, of (i) the normal to the original polished surface and the tilted surfaces of pl&es A. B, C and D (half filled circles); (ii) the normals to the habit planes of the four plates, (unfilled circles); (iii) the { 110)~ planes associated with the sub- bands in each plate; (iv) the orthorhombic axes of the two orientations in each of the four plates A, B, C and n. The unfilled, half filled and filled squares represent’
n’, b and c axes, respectively.
(b) Experimental results
The results of the habit-plane determinations on the
four plates are shown in Figs. 11 and 12. Figure 11
shows the orientation of the plates relative to the
axes of the original cubic crystal and in Fig. 1% the
normals to the four planes have all been plotted in the
same stereographic triangle. The main source of
uncertainty in the habit plane determination arises
from the non-parallelism of the sides of the plates.
Calculations of the extreme positions, using first the
outside edges and then the inside edges, showed that
this could cause an error of between 49.75 and $1.5”
in the direction at right angles to the [OOl],-[Oil],
boundary, but in the direction parallel to this
boundary the error would be negligible except in the
case of plate C where it could amount to &0.25’.
Other sources of error are the errors in measurement
of the traces, &0.25”, and the error in the orientation
determination less than 0.5”. It can be seen from
Fig. 12 that the four habit planes are variants of a
single plane to within these limits of accuracy.
It was found that the plane of the sub-bands within
each plate was within 3“ of a (110) plane of the
original cubic crystal. The relevant {llO},< planes are
indicated in Fig. 11; for each plate the normal to the
relevant {IlO},. plane lies in tha,t stereographic
4 8’ 47O 46’ 4 4” 43’ I I 42’ 41° I -40ZF
I I I I
PO!52 I.051 I.050
I.054
e’1.059 I.056
m
oW **
0” D’
FIG. 12. Stereographic projection showing the varia- tions of habit plane with P for the (TV+, w -) solution, and the measured habit planes (circles). Only the circles marked B and C are in their measured positions; the other poles have been transferred by symmetry
operations to the stereographic triangles shown.
SMITH AND BOWLES: TRANSFORMATION IN Au(:u 411
(b)
FIG. 13. Stereographic projections showing the measured orientations of the twenty-four (lOO>o axes (crosses) relative to the (100)~ axes. The rectangles indicate the estimated error of these determinations. The orienta-
tions predicted by the theory are also shown (dots).
triangle which has a (loo),-( 11 l), boundary, common
with the stereographic triangle containing the normal
to the habit plane of the plate.
The results of the orientation determinations are
shown in Figs. 11 and 13. Each of the four plates
was found to contain two twin orthorhombic
orientations making a total of eight orientations in all.
Fig. 13 shows the orientations of the 24 (lOO), axes
relative to the (loo), axes and indicates the accuracy
of the measurement. Fig. 11 shows the relative
orientations of the orthorhombic axes and the four
habit planes, and also shows which orientations were
found in each plate. Inspection of this diagram shows
that the twinning plane relating the two orientations
in each plate is a (1Ol)o plane and that in each case
this plane coincides with the plane of the sub-bands
in that plate.
The angles through which the original polished
surface was tilted by the production of the four plates
were found to be equal within the accuracy of the
412 ACTA METALLURGICA, VOL. 8, 1960
TABLE 1. Analysis of surface tilts produced by plates A, B, C and D on an orimel surface taken
as (0.07846, 0.61568, 0.78409)c
I Orientation of tilted surfrtces
Plate
A
B
c II
Measured
(0.08083, 0.65252, 0.75345)~ (0.07992. 0.57716. 0.81271)~ iO.07707; 0.65274; 0.75365jc (0.07518, 0.57745, 0.81295)c
Predicted
(0.08045, 0.65261, 0.75341)~ (0.07955. 0.57735. 0.812611~ iO.07690; 0.65258; 0.75381jc (0.07599, 0.57774, 0.81267)~
The predicted angle of tilt is in rll cases 2O 45’ * 1’ in agreement with the measured value 2’ 45’.
method used. The measured angle was 2” 45’. Plates
A and C were tilted in (approximately) opposite
senses to plates B and D. The normals to the original
and tilted surfaces are shown in Fig. 11 and unit
vectors in the directions of these normals are given
in Table 1.
The lattice parameter, a, of the face-centred cubic
phase was found to be 3.872 A hO.002. Using this
value to calibrate the composite film (Section 3), the
values obtained for the axial ratios-b/a’ and c/a’ of the
orthorhombic pseudo-cell-were 0.9960 +0.0003 and
0.9244 &0.0003, respectively. The value obtained for
the ratio a’/a of the orthorhombic to the cubic lattice
parameter, was 1.0276 fO.0003. The parameters of
the orthorhombic pseudo cell were thus a’ = 3.976 A,
b = 3.963 A and c = 3.678 A.
4. COMPARISON OF THEORY AND EXPERIMENT
The phenomenological theory of martensitic trans-
formations was developed from an analysis of the
geometrical features associated with the formation of
martensite plates within a parent crystal. The present
investigation has shown that the low temperature
transformation to the tetragonal structure in Au-&
does not occur by the formation of plates of the new
phase within the old. Hence, it does not conform with
the characteristics on which the phenomenological
theory is based and it is not appropriate to attempt to
apply the theory to this transformation. The trans-
formation to the orthorhombic phase, on the other
hand, exhibits the same kind of geometrical features
as martensitic transformations and it is therefore
appropriate to see whether this transformation
conforms with the theory.
The data required for substitution into the equations
of the theory are the correspondence between the
original and final lattices, the operative twinning
plane and the axial ratios of the orthorhombic lattice.
Considering the variant B in Fig. 11 it is evident that
the correspondence for this variant is given by the
relation*
[o ; 91 = I[c ; xl
where I is the identity matrix.
The operative twinning planes associated with the
various orthorhombic orientations were identified by
the sub-band analysis and this identification was
confirmed by the observation that the twinning plane
of the twin orientations found in each plate was the
same as the plane of the sub-bands in that plate.
Thus, from Fig. 11, the plane from which the twinning
plane in variant B was generated may be identified
as the plane (101)o. The axial ratios of the ortho-
rhombic lattice have been given in the preceding
section.
The predictions of the theory may now be calculated
by substituting these data into the equations derived
in Ref. (3). The theory involves a variable parameter,
6 = 6a’/a, where a and a’ are the lattice parameters of
the cubic and orthorhombic phases respectively, and
6 defines a small unknown dilation. For any given
value of 8 the theory yields four solutions for the
habit plane, these being simply variants of the same
plane. The variation of the habit plane with 0s
predicted by using the (tc+, o-) solution,‘3) is
shown in Fig. 12. The variable parameter e2 is
evaluated by finding that value which gives the best
agreement with the observed habit plane. It can be
seen from Fig. 12 that the predicted habit plane is in
good agreement with the observed habit plane, B,
when e2 = 1.053, and this value has been used in
deriving the predictions described below. Since the
measured value of alla is 1.076, it follows that
6 = 0.9986. It should be noted that although the
predicted habit plane versus e2 curve also passes close
to the experimental habit plane, C, it is not permissible
to evaluate e2 to give agreement with this habit plane,
since the generation of the orientations in plate C
involves a correspondence and twinning plane
different from those used in deriving the habit plane
versus e2 curve. It should also be noted that if 6 is
assumed to be exactly unity,(334) in which case
8s = (at/a)” = 1.056, the predicted habit plane is then
about three degrees from the observed plane and in the
neighbouring stereographic triangle.
Of the four strain matrices that are obtained on
substitution in the equations of Ref. (3), only one defines directly any of the eight orientations in the
four plates studied. This solution, the (a+, w-)
* The notation used in this paper is the same &s that used in (2) and (3). The basis symbols o a,nd c refer to the ortho- rhombic and cubic bases respectively, the metric (oGo) of the orthorhombic basis being (diag. (a’)*. b*, 2).
SMITH AND BOWLES: TRANSFORMATION IN AuCu 413
FIG. 14. Stereographic projection showing the data of Fig. 13 reduced by symmetry operations to a group of five poles around one (100)~ axis. The shaded areas are the areas common to the separate determinations of each pole. The orientations predicted for Be = 1.053, 6 = 0.9986 and P = 1.056, 6 = 1 are shown as filled
circles and open circles, respectively.
solution, describes the B orientation directly. This
result is expected because the B orientation is the only
one of the eight orientations present for which the
above choice of correspondence and twinning plane
is appropriate. Of the other three solutions, one
describes an orientation that is a twin of a variant of
the B orientation, and the other two describe variants
of these two orientations.
The invariant line strain for the (a+, co-) solution is
1.025815, 0.001258,
(cSC) = -0.000620, 1.021767,
-0.026548, 0.024773,
Since the correspondence is described by the
identity matrix, it follows that the successive columns
of this strain matrix are vectors describing the
orientation relative to the basis c of the axes [lOO],,
[OlO], and [OOl], respectively. These predicted directions are plotted in Fig. 13 for comparison with
the measured orientation relationship. The pre-
dictions for the other seven orientations are also
shown in Fig. 13. The predicted orientations A, C and
D are simply variants of the predicted orientation B and have been derived therefrom by performing the
symmetry operations that produce the required
changes of habit plane. The twin orientations A,, B,, C, and D,, have been derived from the (a-, w-)
solution in a similar manner. It can be seen from
Fig. 13 that in all cases the agreement be&Teen the
predicted and measured orientation is very good.
In Fig. 14 the orientations predicted for e2 = 1.053
and for e2 = 1.056 have been plotted for comparison
with the experimental data. For this purpose, the
experimental observations have been reduced by
means of the appropriate symmetry operations to a
group of five poles around one (loo), axis. It can be
seen from this presentation of the results that the
predictions for 8s = 1.053 are systematically in better
agreement than those for f32 = 1.056.
The invariant plane strain, describing the shape
change accompanying the transformation, was found
by substitution into the equations of Ref. (3) to be
(cP,o) = I + m dp’
414 ACTA METALLURGICA, VOL. 8, 1960
where
d = [0.038122, 0.659595; 0.750654],,
p’ = (-0.037122, 0.677769, -0.734336), and
m= 0.048692.
The surface tilts produced by each of the four
plates A, B, C and D were calculated using the above
strain and the appropriate variants of the original
surface. The calculated angles of tilt for plates A, B,
C and D were 2” 45’, 2” 44’, 2’ 44’ and 2’ 44’, re-
spectively, in excellent agreement with the measured
angle of 2” 45’ &5’. In each case the predicted tilt
was in the observed sense, Table 1.
5. DISCUSSION
The results of the present investigation not only
establish that the diffusion controlled cubic to ortho-
rhombic transformation in AuCu conforms with the
phenomenological theory of martensitic transfor-
mations, but they also provide a more critical test of
the theory than has previously been made, The
agreement between the predicted and observed habit
planes is of particular interest in this respect. Hitherto,
the habit planes in AuCu and in the various other
systems such as indium-thallium, in which cubic to
tetragonal transformations occur, were thought to be
exactly parallel to {llO}, planes. This caused some
difficulty, for the theory never predicted a {llO},
habit but always an irrational plane close to {llO},.
The present work removes this difficulty for not only
has the existence of the irrational habit plane been
demonstrated, but it has also been shown that the
lateral growth of such plates can produce compound
plates having a pseudo habit plane, {llo},. It now
seems certain that the (11O)o habits that have been
observed in indium-thallium and other similar systems
are also pseudo habit planes. In these cases the axial
ratios are much nearer to unity than in AuCu and
consequently the true habit planes are much closer
to (llO), so that it would be difficult to distinguish
experimentally between the two. It should be noted
that it is not permissible to regard the {llO}, junction
plane as the habit plane for to do so would not be
consistent with one of the original postulates from
which the theory was developed, viz. that no line in
the habit plane is rotated by the transformation.
The midrib of the compound plate is not an unrotated
line but is composed of lengths rotated in opposite
senses to give a saw tooth effect.
With regard to the proposed dilation, the predictions
of the habit plane and orientation relationship have
been shown to be systematically in better agreement
with the experimental results when e2 is chosen to
have the value 1.053 (6 = 0.9986): than for the case
e2 = (a’/c~)~ = 1.056, which corresponds to 6 = 1.
In the case of the orientation relationship, the
predictions for both values of e2 are probably within
the experimental error but the habit plane results
indicate a more definite decision in favour of
o2 = 1.053. Even the most unfavourable combination
of errors does not eliminate the discrepancy between
the habit plane predicted for 6 = 1 and the experi-
mental result. The habit plane predicted for
(~,‘/a)~ = 1.0273, b/a’ = 0.9957, c/a’ = 0.9241, i.e. the
worst combination of parametric errors, lies on the
curve in Fig. 12 exactly one degree to the right of the
point, f32 = 1.056. This is still outside the experi-
mental error, &0.75’, of even the most favourably
situated plate, A.
From one point of view it is not particularly
surprising to find that the crystallographic features of
this transformation conform with the phenomeno-
logical theory of martensitic transformations. As
Bilby(14) has pointed out, the kind of analysis which
has been applied to martensite crystallography should
be applicable to any transformation that produces
relief effects corresponding to a homogeneous strain;
this is because the shape change must then be approxi-
mately an invariant plane strain. However, if as is
widely believed, the crystallographic features of
martensitic transformations are a consequence of a
special kind of transformation mechanism, then it is
very interesting to find the same crystallography, and
hence presumably the same kind of mechanism, in
AuCu where the transformation is accompanied by
place exchanges between neighbouring atoms.
The view is widely held that a transformation gives
rise to relief effects when the interface between the
phases is such that it can maintain its structural
identity during migration. The kind of interface that
is envisaged is an array of dislocations which, as it
moves, produces both the growth of the new structure
and the displacements of the parent phase that
generate the relief. Whilst there can be little doubt
that the growth of the orthorhombic plates in AuCu
proceeds in such a fashion it is not clear at what stage
of growth the martensitic type of interface is estab-
lished. A number of workers in the field of martensitic
nucleation favour the idea that martensite plates
develop from faulted regions produced in the meta-
stable parent phase by dissociation of dislocations.
These faulted regions could grow by displacement of
the imperfect dislocations surrounding them but, in
general, their structure cannot be exactly that of the new phase; as Bilby(15) has shown, the most general
lattice strain that can be achieved by dislocation
SMITH 4ND BOWLES: TRANSFORMATION Ii% AuCu 415
dissociation is an invariant plane strain. Thus to
create a martensite nucleus with the required interface,
further deformation of the faulted region is needed.
This kind of mechanism is at variance with that
proposed by Newkirk et al. u6-16). From an analysis of
the diffuse diffraction effects in the alloy CoPt these
workers proposed that the ordered phase forms as
platelets coherent with the (110) planes of the matrix.
They proposed that these platelets develop into plates
of microscopic dimensions, the lattice strain produced
by the coherence being relieved by self deformation.
The orientation relationship implied by this proposal,
(110),~~) (IOl),; (OIO),Y)) (OOl), and the habit plane
{110), are clearly incompatible with the present
results. Thus a.lthough Newkirk et al. believed that
their mechanism would also apply to the ordering of
AuCu it is now clear that this cannot be true, at least
for the transformation to the orthorhombic phase.
With regard to diffuse diffraction effects the position
is that these studies have never been carried out on
AuCu specimens quenched directly to an ageing
temperature at which the orthorhombic phase forms
so that it is not known whether the early stages of the
formation of orthorhombic plates give rise to such
effects. However, diffuse diffraction might be expected
from thin plates produced by dissociation of dis-
locations.
The present results for the low temperature cubic
to tetragonal transformation are consistent, however,
with the mechanism of Newkirk et al. and in particular
are consistent with the view that the relief effect in
this transformation are attributable to self deformation
by {lOl} mechanical twinning of the tetragonal
product. This is a little surprising for such twinning,
if it occurred in the ordered structure, would change
the orientation of the c axis relative to the layers of
copper and gold atoms, unless it was accompanied by
a re-arrangement of the atoms. From the observation
that these relief effects are faint at first but become
progressively sharper, it seems that twinning occurs
first at a very early stage of ordering and that ad-
ditional shearing takes place as the degree of order in
the twinned portions increases.
Published work on the alloys CoPt and FePt, both
of which possess superlattices, does not exclude the
possibility that in these cases a tetragonal product
could be produced by a mechanism similar to that
operating in the case of orthorhombic AuCu. It
should also be noted that the present work has not
established that the high temperature mode of trans-
formation in AuCu always yields an orthorhombic
product. The possibility remains that a tetragonal
phase may be produced in this manner at some
temperatures.
ACKNOWLEDGMENTS
The authors wish to thank Dr. J. K. Mackenzie for
many helpful discussions in the course of this work
and Dr. W. Boas and Dr. L. M. Clarebrough for their
helpful criticisms of the manuscript.
1.
2.
3.
4.
5.
6.
7. 8.
9.
10.
11.
12.
13. 14.
15. 16.
17.
18.
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